There's no suggestion that the Balrog of Moria was serving Sauron or anyone else save itself.
As for the Mouth- he's merely a mortal Man who had learned sorcery. Since Tolkien himself says the Nine are Sauron's most powerful servants, and the W-K the most potent of those, who are we to disagree?
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The entire plot of The Lord of the Rings could be said to turn on what Sauron didn’t know, and when he didn’t know it.
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